NDLI: Coincidences of Centers of Edge-Incentric, or Balloon, Simplices

Content Provider	Springer Nature Link
Author	Hajja, Mowaffaq
Copyright Year	2006
Abstract	An edge-incentric d-simplex is defined to be a d-simplex S which admits a (d − 1)-sphere that touches all the edges of S internally. The center of such a sphere is called the edge-incenter of S and is denoted by $${\mathcal{\mathcal{B}}}$$ . Equivalently, S is edge-incentric if and only if its vertices are the centers of d + 1 (d − 1)-spheres in mutual external touch, and for this reason one may call such an S a balloon d-simplex. An orthocentric d-simplex is a d-simplex in which the altitudes are concurrent. The point of concurrence is called the orthocenter and is denoted by $${\mathcal{\mathcal{H}}}$$ . The spaces of edge-incentric and of orthocentric d-simplices have the same dimension d in the sense that a d-simplex in either space can be parametrized, up to shape, by d numbers. Edge-incentric and orthocentric tetrahedra are the first two of the four special classes of tetrahedra studied in [1, Chapter IX.B, pp. 294–333].The degree of regularity implied by the coincidence of two or more centers of a general d-simplex is investigated in [8], where it is shown that the coincidence of the centroid $${\mathcal{\mathcal{G}}}$$ , the circumcenter $${\mathcal{\mathcal{C}}}$$ , and the incenter $${\mathcal{\mathcal{I}}}$$ does not imply much regularity. For an orthocentric d-simplex S, however, it is proved in [9] that if any two of the centers $${\mathcal{\mathcal{G}}},{\mathcal{\mathcal{C}}},{\mathcal{\mathcal{I}}}$$ , and $${\mathcal{\mathcal{H}}}$$ coincide, then S is regular. In this paper, the same question is addressed for edge-incentric d-simplices. Among other things, it is proved that if any three of the centers $${\mathcal{\mathcal{G}}},{\mathcal{\mathcal{C}}},{\mathcal{\mathcal{T}}}$$ , and $${\mathcal{\mathcal{B}}}$$ of an edge-incentric d-simplex S coincide, then S is regular, and it is also shown that none of the coincidences $${\mathcal{\mathcal{G}}}\, = \,{\mathcal{\mathcal{B}}},{\mathcal{\mathcal{I}}}\, = \,{\mathcal{\mathcal{B}}} $$ , and $${\mathcal{\mathcal{I}}}\, = \,{\mathcal{\mathcal{G}}}$$ implies regularity (except when d ≤ 3, d ≤ 4, and d ≤ 6, respectively). In contrast with the afore-mentioned results for orthocentric d-simplices, this emphasizes once more the feeling that, regarding many important properties, orthocentric d-simplices are the true generalizations of triangles.Several open questions are posed.
Ending Page	263
Page Count	27
Starting Page	237
File Format	PDF
ISSN	14226383
e-ISSN	14209012
Journal	Results in Mathematics
Issue Number	3
Volume Number	49
Language	English
Publisher	Birkhäuser-Verlag
Publisher Date	2006-12-05
Publisher Place	Basel
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	incenter Balloon simplex touching spheres equivariant simplex touching hyperspheres Symmetry properties of polytopes Mathematics circumcenter edge-incentric simplex equiareal simplex Gram matrix $n$-dimensional polytopes barycentric coordinates centroid equiradial simplex Special polytopes (linear programming, centrally symmetric, etc.) Descartes’ circles theorem regular simplex Soddy spheres equifacetal simplex edge-incenter kite orthocentric simplex
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics

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